Elaiw et al. Boundary Value Problems 2012, 2012:26
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RESEARCH
Open Access
Effect of variable viscosity on vortex instability of
non-Darcy free convection boundary layer flow
adjacent to a non-isothermal horizontal surface in
a porous medium
Ahmed M Elaiw1,4*, Ahmed A Bakr2,4 and Fuad S Ibrahim3,5
* Correspondence:
[email protected]
1
Department of Mathematics,
Faculty of Science, King Abdulaziz
University, P.O. Box 80203, Jeddah
21589, Saudi Arabia
Full list of author information is
available at the end of the article
Abstract
In this article, we study the effect of variable viscosity on the flow and vortex
instability of non-Darcian free convection boundary layer flow on a horizontal surface
in a saturated porous medium. The wall temperature is a power function of the
distance from the origin. The variation of viscosity is expressed as an exponential
function of temperature. The transformed boundary layer equations, which are
developed using a non similar solution approach, are solved by means of a finite
difference method. The analysis of the disturbance flow is based on linear stability
theory. The local Nusselt number, critical Rayleigh number and the associated wave
number at the onset of vortex instability are presented over a wide range of wall to
ambient viscosity ratio parameters μ*= μw/μ∞. The variable viscosity effect is found to
enhance the heat transfer rate and destabilize the flow for liquid heating, while the
opposite trend is true for gas heating.
Keywords: vortex instability, porous media, variable viscosity, non-Darcy, free convection, finite difference method
1 Introduction
The problems of the vortex mode of instability in free and mixed convection flows
over horizontal and inclined heated surfaces in saturated porous media have received
considerable attention (see [1-11]). The instability mechanism is due to the presence of
a buoyancy force component in the direction normal to the surface. The importance of
the problems is due to a large number of technical applications, such as material transfer associated with the storage of radioactive nuclear waste, separation processes in the
chemical industry, filtration, transpiration cooling, transport processes in aquifers, and
ground water pollution. Some researchers considered Darcy model [1-5], others considered non-Darcy model [6-11]. A comprehensive literature survey on this subject can
be found in the recent book by Nield and Bejan [12].
All of the studies mentioned previously dealt with constant viscosity. The fundamental analysis of convection through porous media with temperature dependent viscosity
is driven by several contemporary engineering applications from the cooling of electronic devices to porous journal bearings and is important for studying the variations in
© 2012 Elaiw et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
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provided the original work is properly cited.
Elaiw et al. Boundary Value Problems 2012, 2012:26
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constitutive properties. The effect of variable viscosity for convective heat transfer
through porous media has also been widely studied (e.g., [13-19]). The effect of variations in viscosity on the instability of flow and temperature fields are discussed by
Kassoy and Zebib [13] and Gray et al. [14]. Lai and Kulacki [15] considered the variable viscosity effect for mixed convection along a vertical plate embedded in a saturated porous medium. The effect of variable viscosity on non-Darcy, free or mixed
convection flow on a horizontal surface in a saturated porous medium is studied by
Kumari [16]. Jayanthi and Kumari [17] studied the same problem presented in Kumari
[16] for vertical surface. Afify [18] studied the effects of non-Darcy, variable viscosity
and Hartmann-Darcy number on free convective heat transfer from an impermeable
vertical plate embedded in a thermally stratified, fluid saturated porous medium for
the case of power-law variation in the wall temperature. In [15-18], the viscosity of the
fluid is assumed to vary as an inverse linear function of temperature, whereas in [19],
the variation of viscosity with temperature is represented by an exponential function.
The effects of variable viscosity on the vortex instability of horizontal free and mixed
convection boundary layer flows in a saturated porous medium for isothermal and
non-isothermal surfaces are studied by Jang and Leu [19] and Elaiw et al. [20], respectively. However, the effect of variable viscosity on the flow and vortex instability of
non-Darcian free convection boundary layer flow over a non-isothermal horizontal
plate does not seem to have been investigated. This motivated the present
investigation.
The purpose of this article is to examine in details the effect of temperature-dependent
viscosity on the flow and vortex instability of a horizontal non-Darcian free convection
boundary layer flow in a saturated porous medium with variable wall temperature. This
is accomplished by considering the Ergun model equation of motion. The variation of
viscosity with temperature is represented by an exponential function, which is more
accurate than linear function especially for large temperature differences. The transformed boundary layer equations, which are given using a non similar solution
approach, are solved by means of a finite difference method. The stability analysis is
based on the linear stability theory. The resulting eigenvalue problem is solved using a
finite difference scheme.
2 Analysis
2.1 The main flow
We consider a semi-infinite non-isothermal horizontal surface (Tw) embedded in a
porous medium (T∞) as shown in Figure 1, where x represents the distance along the
plate from its leading edge, and y the distance normal to the surface. The wall temperature is assumed to be a power function of x, i.e., Tw(x) = T∞ + Axl, where A is a
constant and l is the parameter representing the variation of the wall temperature. In
order to study transport through high porosity media, the original Darcy model is
improved by including inertia. For the mathematical analysis of the problem we
assume that, (i) local thermal equilibrium exists between the fluid and the solid phase,
(ii) the physical properties are constant, except for the viscosity μ and the density r in
the bouncy force, (iii) Ergun non-Darcy model [21] is considered, and (iv) the Boussinesq approximation is valid. With these assumptions, the governing equations are
given by:
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Figure 1 The physical model and coordinate system.
∂u ∂v
+
= 0,
∂x ∂y
u+
K∗ 2
K ∂P
u =−
,
v
μ ∂x
K∗ 2
K
v+
v =−
v
μ
u
(1)
∂P
+ ρg ,
∂y
∂T
∂T
∂2T
+v
=α 2,
∂x
∂y
∂y
(2)
(3)
(4)
where r = r∞[1 - b(T - T∞)] is the fluid density. Note that the second term on the
left-hand side of Equations (2) and (3) represents the inertia force, where K* is the
inertia coefficient in Ergun model. As K* = 0, Equations (2) and (3) reduce to Darcy
model.
The viscosity of the fluid μ is assumed to vary with temperature according to the following exponential form
μ = μ∞ e
T−T
A1 T −T∞
w
∞
,
(5)
where μ ∞ is the absolute viscosity at ambient temperature and A 1 is a constant
adopted from the least square fitting for a particular fluid. Formula (5) is a generalization of that used in [19], where the wall temperature is taken to be constant.
The pressure terms appearing in Equations (2) and (3) can be eliminated through
cross-differentiation. The boundary layer assumption yields ∂/∂x ≪ ∂/∂y and v ≪ u.
With ψ being a stream function such that u = ∂ψ/∂y and v = -∂ψ/∂x, the Equations
(1)-(4) become
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∂ψ ∂ 2 ψ
∂μ
∂T
μ + 2ρ∞ K ∗
+u
= −Kρ∞ gβ ,
2
∂y ∂y
∂y
∂x
(6)
∂ψ ∂T
∂ψ ∂T
∂2T
−
=α 2.
∂y ∂x
∂x ∂y
∂y
(7)
The boundary conditions are defined as follows
∂ψ
= 0, T(x, 0) = Tw = T∞ + Axλ ,
∂x
u(x, ∞) = 0,
T(x, ∞) = T∞ .
v(x, 0) = −
(8)
By introducing the following similarity variables:
y 1/3
ψ(x, y)
η(x, y) = Rax ,
f (ξ , η) =
,
1/3
x
αRax
T − T∞
x (2λ−1)/3
θ (ξ , η) =
,ξ =
,
Tw − T∞
d
(9)
Equations (5)-(7) become
A1
μ
=e
μ∞
T−T∞
Tw −T∞
= (μ∗ )θ ,
(10)
λ−2
2λ − 1 ∂θ
2/3
(1 + 2ErRad ξ (μ∗ )−θ f ) f = −(ln μ∗ )f θ − (μ∗ )−θ λθ +
ηθ +
ξ
,
3
3
∂ξ
(11)
2λ − 1 ∂θ
λ+1
2λ − 1 ∂f
θ = λθ +
ξ
f −
f+
ξ
θ ,
3
∂ξ
3
3
∂ξ
(12)
where μ∗ = μw μ∞ = eA1 is the wall to ambient viscosity ratio parameter,
Rax =
ρ∞ gβK(Tw −T∞ )x
αμ∞
is the local Rayleigh number, Rad =
number based on the pore diameter and Er =
K∗α
dν∞
ρ∞ gβKAdλ+1
αμ∞
is the Rayleigh
is the Ergun number.
The boundary conditions are transformed as follows:
f (ξ , 0) = 0, θ (ξ , 0) = 1,
f (ξ , ∞) = 0, θ (ξ , ∞) = 0,
(13)
where the primes denote the derivatives with respect to h. The velocity components
and the heat transfer coefficient in term of the local Nusselt number are given by:
2/3
u(x, y) =
αRax
f (ξ , η),
x
1/3
αRax
v(x, y) = −
3x
(14)
∂f (ξ , η)
(λ + 1)f (ξ , η) + (λ − 2) ηf (ξ , η) + (2λ − 1) ξ
, (15)
∂ξ
1/3
Nux Rax = −θ (ξ , 0).
(16)
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2.2 The disturbance flow
The standard linear stability method yields the following:
∂u1 ∂v1 ∂w1
+
+
= 0,
∂x
∂y
∂z
(17)
μ0 u1 + μ1 u0 + 2K ∗ ρ∞ u0 u1 = −K
∗
μ0 v1 + μ1 v0 + 2K ρ∞ v0 v1 = −K
μ0 w1 = −K
u0
∂P1
,
∂x
∂P1
− ρ∞ gβT1 ,
∂y
∂P1
,
∂z
(18)
(19)
(20)
2
∂T1
∂T1
∂T0
∂T0
∂ T1 ∂ 2 T1 ∂ 2 T1
.
+
+
+ v0
+ u1
+ v1
=α
∂x
∂y
∂x
∂y
∂x2
∂y2
∂z2
(21)
where the subscripts 0 and 1 signify the mean flow and disturbance components,
respectively.
Following the order of magnitude analysis method described in detail by Hsu et al.
[1], the terms ∂u1/∂x and ∂2T1/∂x2 in Equations (17) and (21) can be neglected. The
omission of ∂u1/∂x in Equation (17) implies the existence of a disturbance stream
function Ψ1, such as
w1 =
∂1
,
∂y
v1 = −
∂1
.
∂z
(22)
Eliminating P1 from Equations (18)-(20) with the aid of (22) leads to
u0
∂μ1
∂ 2 1 ∂1 ∂μ0
∂u1
+ (μ0 + 2K ∗ ρ∞ u0 )
= μ0
+
,
∂z
∂z
∂x∂y
∂y ∂x
−(μ0 + 2K ∗ ρ∞ v0 )
u0
∂μ1
∂ 2 1 ∂μ0 ∂1
∂ 2 1
∂T1
+ v0
+
= μ0
+ Kρ∞ gβ
,
2
∂z
∂z
∂y2
∂y ∂y
∂z
2
∂T1
∂T1
∂T0
∂1 ∂T0
∂ T1 ∂ 2 T1
.
+
+ v0
+ u1
−
=α
∂x
∂y
∂x
∂z ∂y
∂y2
∂z2
(23)
(24)
(25)
As in Hsu et al. [1], we assume that the three-dimensionless disturbances for neutral
stability are of the form
(26)
(1 , u1 , T1 ) = (x, y), ū(x, y), T̄(x, y) eiaz ,
where a is the spanwise periodic wave number. Substituting Equation (26) into Equations (23)-(25) yields
1
u0 T̄
1 ∂2
∗ ∂ ∂θ
∗
ū =
, (27)
+ (ln μ )
− (ln μ )
∗
∗ −θ
ia ∂x∂y
∂y ∂x
Tw − T∞
1 + 2K
ν∞ (μ ) u0
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T̄
2K ∗ ∗ −θ
∂2 2
−a 1 +
(μ ) v0 −iav0 (ln μ∗ )
∂y2
ν∞
Tw − T∞
u0
∂ T̄
∂ T̄
∂T0
∂T0
+ v0
+ ū
− ia
=α
∂x
∂y
∂x
∂y
+ (ln μ∗ )
∂θ ∂
iaKρ∞ gβ ∗ −θ
(μ ) T̄,
=−
∂y ∂y
μ∞
∂ 2 T̄
2
T̄
.
−
a
∂y2
(28)
(29)
Equations (27)-(29) are solved based on the local similarity approximation [22],
wherein the disturbances are assumed to have weak dependence in the streamwise
direction (i.e., ∂/∂x ≪ ∂/∂h).
On applying the following transformations
k=
ax
1/3
Rax
, F(η) =
1/3
iαRax
, (η) =
T̄
,
Tw − T∞
(30)
into Equations (27)-(29) leads to the following:
2/3
2ErRad
∗
2
∗ −θ
F + (ln μ )θ F − k 1 −
ξ (μ ) G1 F
1/3
Rax
k(ln μ∗ )
1/3
+
G1 = −kRax (μ∗ )−θ ,
1/3
Rax
(31)
(ln μ∗ )f G2
2/3
1 + 2ErRad ξ (μ∗ )−θ f 2λ−1
(32)
∗
G2 λ−2
1/3
3 ηF +
3 + (ln μ )G4 F
= kRax θ F,
−
1/3
2/3
kRax
1 + 2ErRad ξ (μ∗ )−θ f + G3 − k2 + λf −
with the boundary conditions
F(0) = F(∞) = (0) = (∞) = 0,
(33)
where G1 - G4 are given by
λ+1
λ−2
2λ − 1 ∂f
f+
ηf +
ξ ,
3
3
3
∂ξ
λ−2
2λ − 1 ∂θ
ηθ +
ξ ,
G2 = λθ +
3
3
∂ξ
λ+1
2λ − 1 ∂f
f+
ξ ,
G3 =
3
3
∂ξ
λ−2
2λ − 1 ∂θ
ηθ +
ξ ,
G4 =
3
3
∂ξ
G1 =
For fixed values of ξ, l, k, Er, Rad, and μ*, the solution F and Θ is an eigenfunction
for the eigenvalue Rax. When μ* = 1, Er = Rad = ξ = 0, Equations (31) and (32) reduce
to those presented in [1] and when μ* = 1, ξ = 1,
∂
∂ξ
= 0 , l = 0.5 they reduce to those
given in [8]. It may also remarked that for l = Er = Rad = ξ = 0, Equations (31) and
(32) reduce to those presented in Jang and Leu [19].
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Page 7 of 13
3 Numerical scheme
In this section, we compute the approximate values of Rax for Equations (31) and (32)
with boundary conditions (33). An implicit finite difference method is used to solve
first the base flow Equations (11) and (12) with boundary conditions (13). The results
are stored for a fixed step size h, which is small enough to make an accurate linear
interpolation between mesh point. The domain is 0 ≤ h ≤ h∞, where h∞ is the edge of
the boundary layer of the basic flow. For a positive integer N, let h = h∞/N and hi =
ih, i = 0, 1, . . . , N. The problem is discretized with standard centered finite differences
of order two, following Usmani [23]. The solution of the eigenvalue problem is
achieved by using the subroutine GVLRG of the IMSL library, see [24].
4 Results and discussion
Numerical results for the local Nusselt number, neutral stability curves, critical Rayleigh numbers and associate wave numbers at the onset of vortex instability for wall to
ambient viscosity ratio parameter μ* ranging from 0.1 to 10 and Ergun number Er ranging from 0.1 to 0.6 are presented. The effect of the non uniform temperature profile
1.4
P*= 0.2, 0.5,1,2.5,5,10
1.2
Nux/Rax1/3
1.0
0.8
0.6
0.4
0.2
O=0.5, Er=0.1, Rad=10
0.0
0
2
4
6
8
[
Figure 2 The local Nusselt number as a function of ξ for selected values of μ*.
P
[
10
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Page 8 of 13
on the wall is also studied, and corresponds to variation of the parameter l from 0.5 to
2. It is known that as the temperature is increased, the gas viscosity increases and the
liquid viscosity decreases [19]. Therefore, for a heated wall, values of μ* >1 corresponds to the case of gas heating, values of μ* <1 corresponds to the case of liquid
heating.
The local Nusselt number as a function ξ for various values of μ*, Er, and l is shown
in Figures 2 and 3. The heat transfer rate is observed to be more than the constantviscosity case for liquids μ* <1 and it has the opposite trend for gases μ* >1. Further, a
lower local Nusselt number is seen to occur with greater values of Er and ξ. Furthermore, a higher Nusselt number is seen to occur with greater values of l.
Figures 4 and 5, respectively, show the neutral stability curves, in terms of Rayleigh
numbers Rax and the dimensionless wave number k for selected values of μ* and ξ.
2.0
Rad =10, P=0.5
Er = 0, 0.1, 0.6
Nux/Rax1/3
1.6
1.2
0.8
O O O 0.4
0
2
4
6
8
[
Figure 3 The local Nusselt number as a function of ξ for selected values of Er and l.
O
[
10
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Page 9 of 13
10000
Rad =10, [=1, O=0.5, Er =0.1
Rax
1000
100
P*= 0.2, 0.5, 1.0, 5, 10
10
0.0
0.5
1.0
1.5
2.0
k
Figure 4 Neutral stability curves for selected values of μ*.
P
Observe that as μ* and ξ increase, the neutral stability curves shift to a higher Rayleigh
number Rax. On the other hand, for liquid heating μ* <1, the neutral stability curves
shift to lower Rayleigh numbers and higher wave numbers, indicating a destabilization
of the flow, whereas for gas heating μ* >1, the opposite trend is true.
The critical Rayleigh number and wave number are plotted as a function of Rad for
selected values of μ* and l in Figures 6 and 7. Observe that, as the variable viscosity
parameter μ* or l increases, the critical Rayleigh Ra∗x increases. Also, as Ra d is
increased the critical Rayleigh number Ra∗x is decreased. Further, the critical wave
number k* is increased as Rad and l are increased and μ* is decreased. From these figures it can be seen that, the inertia effects reduce the heat transfer rate and destabilize
the flow. Finally, we conclude that, the variable viscosity effect enhances the heat transfer rate and destabilizes the flow for liquid heating, while the opposite tend is true for
gas heating.
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Page 10 of 13
1000
Rax
Rad =10, P=1, O=0.5, Er=0.1
100
[=0.1, 1, 5
10
0.0
0.5
1.0
1.5
2.0
k
Figure 5 Neutral stability curves for selected values of ξ.
[
5 Conclusions
The non-Darcy free convection flow on a semi-infinite non-isothermal horizontal plate
embedded in a porous medium with variable viscosity is investigated. The effects of
variable viscosity characterized by the parameter μ* on the flow and vortex instability
are examined. The governing partial differential equations are transformed to a non
similar form by introducing appropriate transformations and are solved numerically
using an implicit finite difference scheme. The resulting eigenvalue problem is solved
by using a finite difference scheme. The effects of all involved parameters on the local
Nusselt number, critical Rayleigh and associated wave number are presented. The
results show that, for liquid heating, the variable viscosity effect enhances the heat
transfer rate and destabilizes flow; for gas heating, the opposite is true.
Nomenclature
a spanwise wave number
d mean particle diameter or pore diameter
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Page 11 of 13
1000
P P P Rax*
Er =0.1, [=1
100
O= 0.5,, 1,, 2
10
0
2
4
6
8
10
Rad
Figure 6 Critical Rayleigh number as a function of Rad for selected values of μ* and l.
P
O
f dimensionless base state stream function
F dimensionless disturbance stream function
g gravitational acceleration
i complex number
k dimensionless wave number
K permeability of porous medium
K* inertial coefficient in Ergun Equation
1.5
Er =0.1, [=1
k*
1
0.5
P P P
O=0.5, 1, 2
0
0
2
4
6
8
10
Ra
Figure 7 Critical wave number as a function of Rad for selected values of μ* and l.
P
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Page 12 of 13
Nux local Nusselt number
P pressure
Rax local Rayleigh number
Rad Rayleigh number based on the pore diameter
T fluid temperature
u, v, w volume averaged velocity in the x, y, z directions
x, y, z axial, normal and spanwise coordinates
Greek symbols
a thermal diffusivity
b volumetric coefficient of thermal expansion
h pseudo-similarity variable
θ dimensionless base state temperature
Θ dimensionless disturbance temperature
μ dynamic viscosity of the fluid
ξ non similarity parameter
l exponent in the wall temperature variation
ν kinematic viscosity
μ* wall to ambient viscosity ratio, μ∗ =
μw
μ∞
ψ stream function
Subscripts
w conditions at the wall
∞ conditions at the free stream
0 basic undisturbed quantities
1 disturbed quantities
Superscripts
* critical value
’ differentiation w.r.t h
Acknowledgements
The authors were grateful to the anonymous reviewers for constructive suggestions and valuable comments, which
had improved the quality of the article.
Author details
1
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi
Arabia 2Department of Mathematics, Faculty of Science for Girls, King Khalid University, Abha, Saudi Arabia
3
Department of Mathematics, University Collage in Makkah, Umm-Alqura University, Makkah, Saudi Arabia
4
Department of Mathematics, Faculty of Science, Al-Azhar University (Assiut Branch), Assiut, Egypt 5Department of
Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
Authors’ contributions
All authors participated in the design of the study and performed the numerical simulation. All authors conceived of
the study, and participated in its design and coordination. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 1 October 2011 Accepted: 27 February 2012 Published: 27 February 2012
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Cite this article as: Elaiw et al.: Effect of variable viscosity on vortex instability of non-Darcy free convection
boundary layer flow adjacent to a non-isothermal horizontal surface in a porous medium. Boundary Value
Problems 2012 2012:26.
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